Complementarity Problems - ampl

17 Pages·2002·0.13 MB·English

19
________________________
________________________________________________________________________
Complementarity Problems
A variety of physical and economic phenomena are most naturally modeled by saying
that certain pairs of inequality constraints must be complementary, in the sense that at
least one must hold with equality. These conditions may in principle be accompanied by
an objective function, but are more commonly used to construct complementarity prob-
lems for which a feasible solution is sought. Indeed, optimization may be viewed as a
special case of complementarity, since the standard optimality conditions for linear and
smooth nonlinear optimization are complementarity problems. Other kinds of comple-
mentarity problems do not arise from optimization, however, or cannot be conveniently
formulated or solved as optimization problems.
TheAMPLoperatorcomplementspermits complementarity conditions to be speci-
fied directly in constraint declarations. Complementarity models can thereby be formu-
lated in a natural way, and instances of such models are easily sent to special solvers for
complementarity problems.
To motivate the syntax of complements, we begin by describing how it would be
used to model a few simple economic equilibrium problems, some equivalent to linear
programs and some not. We then give a general definition of thecomplementsopera-
tor for pairs of inequalities and for more general ‘‘mixed’’ complementarity conditions
via double inequalities. Where appropriate in these sections, we also comment on an
AMPLinterface to thePATHsolver for ‘‘square’’ mixed complementarity problems. In a
final section, we describe how complementarity constraints are accommodated in several
of AMPL’s existing features, including presolve, constraint-name suffixes, and generic
synonyms for constraints.
19.1 Sources of complementarity
Economic equilibria are one of the best-known applications of complementarity con-
ditions. We begin this section by showing how a previous linear programming example
in production economics has an equivalent form as a complementarity model, and how
419
420 COMPLEMENTARITY PROBLEMS CHAPTER 19
________________________________________________________________________
____________________________________________________________________________________________________________________________________________________________________________________
set PROD; # products
set ACT; # activities
param cost {ACT} > 0; # cost per unit of each activity
param demand {PROD} >= 0; # units of demand for each product
param io {PROD,ACT} >= 0; # units of each product from
# 1 unit of each activity
var Level {j in ACT} >= 0;
minimize Total_Cost: sum {j in ACT} cost[j] * Level[j];
subject to Demand {i in PROD}:
sum {j in ACT} io[i,j] * Level[j] >= demand[i];
Figure 19-1: Production cost minimization model (econmin.mod).
________________________________________________________________________
____________________________________________________________________________________________________________________________________________________________________________________
bounded variables are handled though an extension to the concept of complementarity.
We then describe a further extension to price-dependent demands that is not motivated by
optimization or equivalent to any linear program. We conclude by briefly describing
other complementarity models and applications.
A complementarity model of production economics
In Section 2.4 we observed that the form of a diet model also applies to a model of
production economics. The decision variables may be taken as the levels of production
activities, so that the objective is the total production cost,
minimize Total_Cost: sum {j in ACT} cost[j] * Level[j];
where cost[j] and Level[j] are the cost per unit and the level of activity j. The
constraints say that the totals of the product outputs must be at least the product demands:
subject to Demand {i in PROD}:
sum {j in ACT} io[i,j] * Level[j] >= demand[i];
with io[i,j] being the amount of product i produced per unit of activity j, and
demand[i] being the total quantity of product i demanded. Figures 19-1 and 19-2
show this ‘‘economic’’ model and some data for it.
Minimum-cost production levels are easily computed by a linear programming solver:
ampl: model econmin.mod;
ampl: data econ.dat;
ampl: solve;
CPLEX 8.0.0: optimal solution; objective 6808640.553
3 dual simplex iterations (0 in phase I)
SECTION 19.1 SOURCES OF COMPLEMENTARITY 421
________________________________________________________________________
____________________________________________________________________________________________________________________________________________________________________________________
param: ACT: cost :=
P1 2450 P1a 1290
P2 1850 P2a 3700 P2b 2150
P3 2200 P3c 2370
P4 2170 ;
param: PROD: demand :=
AA1 70000
AC1 80000
BC1 90000
BC2 70000
NA2 400000
NA3 800000 ;
param io (tr):
AA1 AC1 BC1 BC2 NA2 NA3 :=
P1 60 20 10 15 938 295
P1a 8 0 20 20 1180 770
P2 8 10 15 10 945 440
P2a 40 40 35 10 278 430
P2b 15 35 15 15 1182 315
P3 70 30 15 15 896 400
P3c 25 40 30 30 1029 370
P4 60 20 15 10 1397 450 ;
Figure 19-2: Data for production models (econ.dat).
________________________________________________________________________
____________________________________________________________________________________________________________________________________________________________________________________
ampl: display Level;
Level [*] :=
P1 0
P1a 1555.3
P2 0
P2a 0
P2b 0
P3 147.465
P3c 1889.4
P4 0
Recall (from Section 12.5) that there are also dual or marginal values — or ‘‘prices’’ —
associated with the constraints:
ampl: display Demand.dual;
Demand.dual [*] :=
AA1 16.7051
AC1 5.44585
BC1 57.818
BC2 0
NA2 0
NA3 0
422 COMPLEMENTARITY PROBLEMS CHAPTER 19
In the conventional linear programming interpretation, the price on constraint i gives,
within a sufficiently small range, the change in the total cost per unit change in the
demand for producti.
Consider now an alternative view of the production economics problem, in which we
define variables Price[i] as well as Level[j] and seek an equilibrium rather than
an optimum solution. There are two requirements that the equilibrium solution must sat-
isfy.
First, for each product, total output must meet demand and the price must be nonnega-
tive, and in addition there must be acomplementarity between these relationships: where
production exceeds demand the price must be zero, or equivalently, where the price is
positive the production must equal the demand. This relationship is expressed in AMPL
by means of thecomplementsoperator:
subject to Pri_Compl {i in PROD}:
Price[i] >= 0 complements
sum {j in ACT} io[i,j] * Level[j] >= demand[i];
When two inequalities are joined by complements, they both must hold, and at least
one must hold with equality. Because our example is indexed over the set PROD, it sets
up a relationship of this kind for each product.
Second, for each activity, there is another relationship that may at first be less obvi-
ous. Consider that, for each unit of activityj, the value of the resulting productioutput
in terms of the model’s prices isPrice[i] * io[i,j]. The total value of all outputs
from one unit of activityjis thus
sum {i in ACT} Price[i] * io[i,j]
At equilibrium prices, this total value cannot exceed the activity’s cost per unit,
cost[j]. Moreover, there is a complementarity between this relationship and the level
of activity j: where cost exceeds total value the activity must be zero, or equivalently,
where the activity is positive the total value must equal the cost. Again this relationship
can be expressed inAMPLwith thecomplementsoperator:
subject to Lev_Compl {j in ACT}:
Level[j] >= 0 complements
sum {i in PROD} Price[i] * io[i,j] <= cost[j];
Here the constraint is indexed overACT, so that we have a complementarity relationship
for each activity.
Putting together the two collections of complementarity constraints, we have the lin-
ear complementarity problem shown in Figure 19-3. The number of variables and the
number of complementarity relationships are equal (to activities plus products), making
this a ‘‘square’’ complementarity problem that is amenable to certain solution techniques,
though not the same techniques as those for linear programs.
Applying thePATHsolver, for example, the complementarity problem can be seen to
have the same solution as the related minimum-cost production problem:
SECTION 19.1 SOURCES OF COMPLEMENTARITY 423
________________________________________________________________________
____________________________________________________________________________________________________________________________________________________________________________________
set PROD; # products
set ACT; # activities
param cost {ACT} > 0; # cost per unit of each activity
param demand {PROD} >= 0; # units of demand for each product
param io {PROD,ACT} >= 0; # units of each product from
# 1 unit of each activity
var Price {i in PROD};
var Level {j in ACT};
subject to Pri_Compl {i in PROD}:
Price[i] >= 0 complements
sum {j in ACT} io[i,j] * Level[j] >= demand[i];
subject to Lev_Compl {j in ACT}:
Level[j] >= 0 complements
sum {i in PROD} Price[i] * io[i,j] <= cost[j];
Figure 19-3: Production equilibrium model (econ.mod).
________________________________________________________________________
____________________________________________________________________________________________________________________________________________________________________________________
ampl: model econ.mod;
ampl: data econ.dat;
ampl: option solver path;
ampl: solve;
Path v4.5: Solution found.
7 iterations (0 for crash); 33 pivots.
20 function, 8 gradient evaluations.
ampl: display sum {j in ACT} cost[j] * Level[j];
sum{j in ACT} cost[j]*Level[j] = 6808640
Further application ofdisplayshows thatLevelis the same as in the production eco-
nomics LP and thatPricetakes the same values thatDemand.dualhas in the LP.
Complementarity for bounded variables
Suppose now that we extend our models by placing bounds on the activity levels:
level_min[j] <= Level[j] <= level_max[j]. The equivalence between the
optimization problem and a square complementarity problem can be maintained, pro-
vided that the complementarity relationship for the activities is generalized to a ‘‘mixed’’
form. Where an activity’s cost is greater than its total value (per unit), the activity’s level
must be at its lower bound (much as before). Where an activity’s level is between its
bounds, its cost must equal its total value. And an activity’s cost may also be less than its
total value, provided that its level is at its upper bound. These three relationships are
summarized by another form of thecomplementsoperator:
subject to Lev_Compl {j in ACT}:
level_min[j] <= Level[j] <= level_max[j] complements
cost[j] - sum {i in PROD} Price[i] * io[i,j];
424 COMPLEMENTARITY PROBLEMS CHAPTER 19
________________________________________________________________________
____________________________________________________________________________________________________________________________________________________________________________________
set PROD; # products
set ACT; # activities
param cost {ACT} > 0; # cost per unit of each activity
param demand {PROD} >= 0; # units of demand for each product
param io {PROD,ACT} >= 0; # units of each product from
# 1 unit of each activity
param level_min {ACT} > 0; # min allowed level for each activity
param level_max {ACT} > 0; # max allowed level for each activity
var Price {i in PROD};
var Level {j in ACT};
subject to Pri_Compl {i in PROD}:
Price[i] >= 0 complements
sum {j in ACT} io[i,j] * Level[j] >= demand[i];
subject to Lev_Compl {j in ACT}:
level_min[j] <= Level[j] <= level_max[j] complements
cost[j] - sum {i in PROD} Price[i] * io[i,j];
Figure 19-4: Bounded version of production equilibrium model (econ2.mod).
________________________________________________________________________
____________________________________________________________________________________________________________________________________________________________________________________
When a double inequality is joined to an expression bycomplements, the inequalities
must hold, and either the expression must be zero, or the lower inequality must hold with
equality and the expression must be nonnegative, or the upper inequality must hold with
equality and the expression must be nonpositive.
A bounded version of our complementarity examples is shown in Figure 19-4. The
PATHsolver can be applied to this model as well:
ampl: model econ2.mod;
ampl: data econ2.dat;
ampl: option solver path;
ampl: solve;
Path v4.5: Solution found.
9 iterations (4 for crash); 8 pivots.
22 function, 10 gradient evaluations.
ampl: display level_min, Level, level_max;
: level_min Level level_max :=
P1 240 240 1000
P1a 270 1000 1000
P2 220 220 1000
P2a 260 680 1000
P2b 200 200 1000
P3 260 260 1000
P3c 220 1000 1000
P4 240 240 1000
;
The results are the same as for the LP that is derived from our previous example (Figure
19-1) by adding the bounds above to the variables.
SECTION 19.1 SOURCES OF COMPLEMENTARITY 425
________________________________________________________________________
____________________________________________________________________________________________________________________________________________________________________________________
set PROD; # products
set ACT; # activities
param cost {ACT} > 0; # cost per unit of each activity
param io {PROD,ACT} >= 0; # units of each product from
# 1 unit of each activity
param demzero {PROD} > 0; # intercept and slope of the demand
param demrate {PROD} >= 0; # as a function of price
var Price {i in PROD};
var Level {j in ACT};
subject to Pri_Compl {i in PROD}:
Price[i] >= 0 complements
sum {j in ACT} io[i,j] * Level[j]
>= demzero[i] - demrate[i] * Price[i];
subject to Lev_Compl {j in ACT}:
Level[j] >= 0 complements
sum {i in PROD} Price[i] * io[i,j] <= cost[j];
Figure 19-5: Price-dependent demands (econnl.mod).
________________________________________________________________________
____________________________________________________________________________________________________________________________________________________________________________________
Complementarity for price-dependent demands
If complementarity problems only arose from linear programs, they would be of very
limited interest. The idea of an economic equilibrium can be generalized, however, to
problems that have no LP equivalents. Rather than taking the demands to be fixed, for
example, it makes sense to view the demand for each product as a decreasing function of
its price.
The simplest case is a decreasing linear demand, which could be expressed in AMPL
as
demzero[i] - demrate[i] * Price[i]
where demzero[i] and demrate[i] are nonnegative parameters. The resulting
complementarity problem simply substitutes this expression fordemand[i], as seen in
Figure 19-5. The complementarity problem remains square, and can still be solved by
PATH, but with clearly different results:
ampl: model econnl.mod;
ampl: data econnl.dat;
ampl: option solver path;
ampl: solve;
Path v4.5: Solution found.
11 iterations (3 for crash); 11 pivots.
12 function, 12 gradient evaluations.
426 COMPLEMENTARITY PROBLEMS CHAPTER 19
ampl: display Level;
Level [*] :=
P1 240
P1a 710.156
P2 220
P2a 260
P2b 200
P3 260
P3c 939.063
P4 240
;
The balance between demands and prices now tends to push down the equilibrium pro-
duction levels.
Because thePrice[i]variables appear on both sides of thecomplementsopera-
tor in this model, there is no equivalent linear program. There does exist an equivalent
nonlinear optimization model, but it is not as easy to derive and may be harder to solve as
well.
Other complementarity models and applications
This basic example can be extended to considerably more complex models of eco-
nomic equilibrium. The activity and price variables and their corresponding complemen-
tarity constraints can be comprised of several indexed collections each, and both the cost
and price functions can be nonlinear. A solver such as PATH handles all of these exten-
sions, so long as the problem remains square in the sense of having equal numbers of
variables and complementarity constraints (or being easily converted to such a form as
explained in the next section).
More ambitious models may add an objective function and may mix equality, inequal-
ity and complementarity constraints in arbitrary numbers. Solution techniques for these
so-calledMPECs — mathematical programs with equilibrium constraints — are at a rela-
tively experimental stage, however.
Complementarity problems also arise in physical systems, where they can serve as
models of equilibrium conditions between forces. A complementarity constraint may
represent a discretization of the relationship between two objects, for example. The rela-
tionship on one side of the complements operator may hold with equality at points
where the objects are in contact, while the relationship on the other side holds with equal-
ity where they do not touch.
Game theory provides another class of examples. The Nash equilibrium for a bi-
matrix game is characterized by complementarity conditions, for example, in which the
variables are the probabilities with which the two players make their available moves.
For each move, either its probability is zero, or a related equality holds to insure there is
nothing to be gained by increasing or decreasing its probability.
Surveys that describe a variety of complementarity problems in detail are cited in the
references at the end of this chapter.
SECTION 19.2 FORMS OF COMPLEMENTARITY CONSTRAINTS 427
19.2 Forms of complementarity constraints
AnAMPLcomplementarity constraint consists of two expressions or constraints sepa-
rated by thecomplementsoperator. There are alwaystwo inequalities,whose position
determines how the constraint is interpreted.
If there is one inequality on either side ofcomplements, the constraint has the gen-
eral form
single-inequality complements single-inequality ;
where a single-inequality is any valid ordinary constraint — linear or nonlinear — con-
taining one>=or<=operator. A constraint of this type is satisfied if both of thesingle-
inequalityrelations are satisfied, and at least one is satisfied with equality.
If both inequalities are on the same side of the complements operator, the con-
straint has instead one of the forms
double-inequality complements expression ;
expression complements double-inequality ;
where double-inequality is any ordinary AMPL constraint containing two >= or two <=
operators, andexpressionis any numerical expression. Variables may appear nonlinearly
in either thedouble-inequalityor theexpression(or both). The conditions for a constraint
of this type to be satisfied are as follows:
•if the left side <= or the right side >= of the double-inequality holds with
equality, then theexpressionis greater than or equal to 0;
•if the right side <= or the left side >= of the double-inequality holds with
equality, then theexpressionis less than or equal to 0;
•if neither side of thedouble-inequalityholds with equality, then theexpres-
sionequals 0.
In the special case where the double-inequality has the form 0 <= body <= Infinity,
these conditions reduce to those for complementarity of a pair of single inequalities.
For completeness, the special case in which the left-hand side equals the right-hand
side of the double inequality may be written using one of the forms
equality complements expression ;
expression complements equality ;
A constraint of this kind is equivalent to an ordinary constraint consisting only of the
equality; it places no restrictions on theexpression.
For the use of solvers that require ‘‘square’’ complementarity systems, AMPL con-
verts to square any model instance in which the number of variables equals the number of
complementarity constraints plus the number of equality constraints. There may be any
number of additional inequality constraints, but there must not be any objective. Each
equality is trivially turned into a complementarity condition, as observed above; each
428 COMPLEMENTARITY PROBLEMS CHAPTER 19
added inequality is made complementary to a new, otherwise unused variable, preserving
the squareness of the problem overall.
19.3 Working with complementarity constraints
All ofAMPL’s features for ordinary equalities and inequalities extend in a straightfor-
ward way to complementarity constraints. This section covers extensions in three areas:
expressions for related solution values, effects of presolve and related displays of prob-
lem statistics, and generic synonyms for constraints.
Related solution values
AMPL’s built-in suffixes for values related to a problem and its solution extend to
complementarity constraints, but with two collections of suffixes — of the form
cname.Lsuf and cname.Rsuf — corresponding to the left and right operands of com-
plements, respectively. Thus after econ2.mod (Figure 19-4) has been solved, for
example, we can use the followingdisplaycommand to look at values associated with
the constraintLev_Compl:
ampl: display Lev_Compl.Llb, Lev_Compl.Lbody,
ampl? Lev_Compl.Rbody, Lev_Compl.Rslack;
: Lev_Compl.Llb Lev_Compl.Lbody Lev_Compl.Rbody Lev_Compl.Rslack :=
P1 240 240 1392.86 Infinity
P1a 270 1000 -824.286 Infinity
P2 220 220 264.286 Infinity
P2a 260 680 5.00222e-12 Infinity
P2b 200 200 564.286 Infinity
P3 260 260 614.286 Infinity
P3c 220 1000 -801.429 Infinity
P4 240 240 584.286 Infinity
;
Because the right operand ofLev_Complis an expression, it is treated as a ‘‘constraint’’
with infinite lower and upper bounds, and hence infinite slack.
A suffix of the form cname.slack is also defined for complementarity constraints.
For complementary pairs of single inequalities, it is equal to the lesser of
cname.Lslack and cname.Rslack. Hence it is nonnegative if and only if both
inequalities are satisfied and is zero if the complementarity constraint holds exactly. For
complementary double inequalities of the form
expr complements lbound <= body <= ubound
lbound <= body <= ubound complements expr
cname.slackis defined to be